Finding Characters Satisfying a Maximal Condition for Their Unipotent Support Jay Taylor

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1 Finding Characters Satisfying a Maximal Condition for Their Unipotent Support Jay Taylor Abstract. In this article we extend independent results of Lusztig and Hézard concerning the existence of irreducible characters of finite reductive groups, (defined in good characteristic and arising from simple algebraic groups), satisfying a strong numerical relationship with their unipotent support. Along the way we obtain some results concerning quasi-isolated semisimple elements. 1. Introduction Throughout this article G will be a simple algebraic group over K = F p, an algebraic closure of the finite field F p, where p is a good prime for G. Furthermore, we assume G is defined over F q F p (where q is a power of a p) and F : G G is the associated Frobenius endomorphism. Throughout we will denote an algebraic group in bold and its corresponding fixed point subgroup in roman, for instance G := G F. Let us denote by Cl U (G) the set of all unipotent conjugacy classes of G and by Cl U (G) F all those classes which are F-stable. In [Lus84a, 13.4] and [Lus92, 10] Lusztig defined combinatorially using j- induction and the Springer correspondence a map Φ G : Irr(G) Cl U (G) F. He had previously conjectured in [Lus80] that for any G and any irreducible character χ Irr(G) there should exist a unique class O χ Cl U (G) F of maximal dimension satisfying χ(g) = 0. g Oχ F It was shown in [Lus92], (and later [Gec96]), that in good characteristic O χ always exists and that Φ G (χ) = O χ. We call the class O χ the unipotent support of χ. Recall that Lusztig has shown for each χ Irr(G) there is a well defined integer n χ such that n χ χ(1) is a polynomial in q with integer coefficients. If x G then we write A G (x) for the component group C G (x)/c G (x), furthermore if F(x) = x then we denote again by F the automorphism of A G (x) induced by F. We will denote by A G (x) the quotient group C G (x) F /C G (x) F which may be identified with the fixed point group A G (x) F. In [Lus84a, 13.4] Lusztig gave various properties of the map Φ G which should hold when Z(G) is connected. For our concerns the most important of these are that: Φ G is surjective, n χ divides A G (u) for u O χ and finally that there exists for each O Cl U (G) F at least one χ Φ 1 G (O) such that n χ = A G (u) for u O. This intriguing numerical property has provided several interesting applications to the representation theory of G, namely via Kawanaka s theory of generalised Gelfand Graev representations, (see for instance [Gec99, 3] and [GH08, Theorem 4.5]). Unfortunately, at the time of writing, proving the existence of such characters seems only possible by carrying out a case by case check and the details of this were

2 2 omitted from [Lus84a, 13.4]. A detailed case by case analysis was provided by Hézard in his PhD thesis [Héz04] and also partly by Lusztig in [Lus09]. Note that in the latter reference necessary questions concerning F-stability were not addressed, however results concerning groups with a disconnected centre were considered. It is the main goal of this paper, (using extensively the work of Lusztig and Hézard), to extend in a natural way the existence of characters satisfying n χ = A G (u) for their unipotent support to the case where G has a disconnected centre. Along the way we also restate the results of Hézard so that all simple groups are treated in our paper. Note that the existence of the characters mentioned above will follow from our main result, (Theorem 2.11), as is shown in [Tay13, Theorem 3.1]. In [Tay13] it is also shown that this result gives direct applications to the representation theory of those groups G along the same lines as those obtained by Geck and Geck Hézard. The logical layout of this paper is as follows. In Section 2 we introduce enough notation so that we can accurately state our main result. In Sections 3 and 4 we introduce further notation and conventions that will be required for the case by case check. Section 5 is dedicated to dealing with clarifications required concerning degenerate elements in half-spin groups. In particular we describe explicitly the Springer correspondence for degenerate elements, which may be of independent interest. In Section 6 we recall results of Bonnafé concerning quasi-isolated semisimple elements and prove the existence of F-stable classes. Finally the remaining sections are the execution of the case by case check. Note that we have also included an index of notation for the convenience of the reader. Acknowledgments: The work presented in this article forms the main part of the authors PhD qualification, which was supervised by Prof. Meinolf Geck. The author wishes to express his deepest thanks to Prof. Geck for proposing this problem and for his vital guidance. The author would like to thank Prof. Cédric Bonnafé for many useful discussions and Professors Gunter Malle and Radha Kessar for carefully reading his PhD thesis and providing numerous useful comments. We also thank Prof. Malle for his comments on a preliminary version of this paper. 2. Stating The Result 2.1. Let us fix a regular embedding ι : G G of G into a group with connected centre with the same derived subgroup as G, (see [Lus88, 7]), and denote by ι : G G the induced surjective morphism of dual groups. If G has a connected centre then we simply take G = G and ι to be the identity map. We assume these groups to be fixed and F to be a corresponding Frobenius endomorphism of the dual groups G and G. Assume T 0 B 0 is a maximal torus and Borel subgroup of G, both assumed to be F-stable, then we denote by (W, S) the Coxeter system of G defined with respect to T 0 B 0. Taking T 0 B 0 to be the unique maximal torus and Borel subgroup of G satisfying T 0 = T 0 G and B 0 = B 0 G we have ι naturally identifies (W, S) with the Coxeter system of G defined with respect to T 0 B 0. We now fix F -stable maximal tori T 0 G and T 0 G such that the triples ( G, T 0, F) and ( G, T 0, F ), respectively (G, T 0, F) and (G, T 0, F ), are in duality. We then necessarily have that ι ( T 0 ) = T 0, (see for instance [Tay12, Lemma 1.71]). Denote by (W, T) the Coxeter system of G defined with respect to T 0 B 0 then we have ι naturally identifies (W, T) with the Coxeter system of G defined with respect to T 0 B 0 := ι ( B 0 ). Recall that by duality we have an anti-isomorphism W W denoted by w w which satisfies S = T.

3 For each s T 0 we denote by W (s) W (s) W the Weyl group of the connected reductive group C G (s) and the group {w W s w = ẇ 1 sẇ = s} of all elements commuting with s, (where ẇ N G (T0 ) is a representative of w). We define similar groups for all semisimple elements s T 0 but in this case we have W ( s) = W ( s) as the centraliser of every semisimple element is connected. Assume the conjugacy class of s is F -stable then there exists w W such that F ( s) = sẇ. Choose a Borel subgroup B( s) of C G ( s) containing T 0 then by [Lus84a, Lemma 1.9(i)] we may assume w is the unique element of minimal length in the right coset ww ( s); this element is characterised by the fact that w normalises B( s). We may now define a Frobenius endomorphism F s of C G ( s) by setting F s (h) := ẇf (h) which stabilises T 0 and B( s) hence induces a Coxeter automorphism of W ( s). Taking B(s) := ι (B( s)) to be a Borel subgroup of C G (s) one similarly has a Frobenius endomorphism Fs of C G (s) inducing a Coxeter automorphism of W (s) and satisfying Fs ι = ι F s. We will similarly denote by W( s), W(s) and W(s) the corresponding subgroups of W under the anti-isomorphism between W and W By the Jordan decomposition of characters and [DM91, Proposition 13.20] we have a bijection Ψ s : E( G, s) E(C G ( s), 1) E(C G (s), 1) (2.4) where the first set is the geometric Lusztig series of G determined by s T 0 and the latter sets are the sets of unipotent characters. Note that here we denote by C G ( s), (resp. C G (s) ), the fixed point subgroup of C G ( s), (resp. C G (s) ), under F s, (resp. F s ). By the classification of unipotent characters given in [Lus84a, 4] the set of unipotent characters E(C G ( s), 1) can further be partitioned in to what are called families; we denote these by F. In fact, we have an injective map F W ( F) Irr(W ( s)) whose image is the set of F s -stable families of irreducible characters of W ( s). By the map in (2.4) we have a bijection F F between the families of unipotent characters of C G ( s) and those of C G (s). With this in mind we denote by T G the set of all pairs ( s, W ( F)) satisfying s T 0 lies in an F -stable conjugacy class and the image of s := ι ( s) under an adjoint quotient of G is a quasi-isolated semisimple element, (for a definition see [Bon05, 1.B]), W ( F) Irr(W ( s)) is a family of characters which is invariant under the induced action of F s. Clearly W acts naturally by conjugation on T G and we denote by T G the orbits under this action Let N G denote the set of all pairs (O, E ) where O Cl U (G) is a unipotent class and E is a Q l - constructible local system on O. We will need the notion of a good pair as introduced by Geck in [Gec99, 4.4]. Recall that the Springer correspondence gives us an embedding Irr(W) N G, (see [Lus84b]), which we denote by ρ (O ρ, E ρ ). With this we may now define a function d : Irr(W) N 0 as follows. Let ρ Irr(W) then we then define d(ρ) to be dim B G u where B G u is the variety of all Borel subgroups of G containing u O ρ. an irreducible character to its b-invariant, (see [GP00, 5.2.2]). We also denote by b : Irr(W) N 0 the function which maps With this invariant we may define the Lusztig MacDonald Spaltenstein induction map j W W where W is any subgroup of W, (see [GP00, 5.2.8]). Proposition 2.6 (Lusztig, [Lus09, Theorem 1.5]). Assume ( s, W ( F)) T G is any pair and let ρ 0 W ( F) be the unique special character (see [GP00, Theorem (b)]) then we have Ind W W( s) (ρ 0) = ρ 0 + a combination of ρ Irr(W) with b( ρ) > b(ρ 0 )

4 4 where ρ 0 Irr(W) satisfies b(ρ 0 ) = b(ρ 0). Furthermore the pair (O ρ 0, E ρ 0 ) corresponding to ρ 0 under the Springer correspondence satisfies E ρ 0 = Ql. Definition 2.7. Recall from [Spa85, 1.1(I)] that we have b(ρ) d(ρ) for all ρ Irr(W), hence in the notation of Proposition 2.6 we have an inequality b( ρ) d( ρ) b(ρ 0 ). With this in mind we say a pair ( s, W ( F)) T G is d-good if the following sharper form of Proposition 2.6 holds Ind W W( s) (ρ 0) = ρ 0 + a combination of ρ Irr(W) with d( ρ) > b(ρ 0 ) Recall that we are interested in the characters of G. These are classified in [Lus88] by understanding the restriction of characters from G to G, in particular we have the following. Assume ψ E( G, s) is an irreducible character of G and ψ := Ψ s ( ψ) is the corresponding unipotent character. The quotient group A G (s) acts on ψ by conjugation and we denote the stabiliser of ψ under this action by Stab AG (s)(ψ), (note here A G (s) is defined with respect to F s ). The main result of [Lus88] states that the restriction Res G G ( ψ) is multiplicity free and contains Stab AG (s)(ψ) number of irreducible constituents. The main result below will be concerned with finding characters ψ such that Stab AG (s)(ψ) is maximal To understand the meaning of the word maximal in the previous sentence we must recall some results from [Tay13] concerning unipotent classes. The embedding ι induces a bijection Cl U (G) Cl U ( G) between the sets of unipotent conjugacy classes and we will implicitly identify each O Cl U (G) with its image ι(o), (similarly for the unipotent elements of G and G). Definition Let O Cl U (G) F then we say a class representative u O F is well chosen if A G (u) F = Z G (u) F A G (u), where Z G(u) is the image of Z(G) in A G (u). By [Tay13, Proposition 2.8] the previous definition makes sense, in particular every class O Cl U (G) F of a simple algebraic group contains a well-chosen class representative and we will assume all class representatives of F-stable unipotent classes are well chosen. With this in hand we may now give the main theorem of this paper. Theorem Assume G is a simple algebraic group, p is a good prime for G and O Cl U (G) F. There exists a pair ( s, W ( F)) T G admitting a unipotent character ψ F satisfying the following properties: (P1) n ψ = A G (u). (P2) Stab AG (s)(ψ) = Z G (u) F. (P3) j W W( s) (ρ) corresponds to (O, Q l) under the Springer correspondence where ρ W ( F) is the unique special character. In particular this ensures that Φ G (Ψ 1 s (ψ)) = O. Furthermore the following conditions hold unless G is a spin/half-spin group and A G (u) is non-abelian: (P4) the pair ( s, W ( F)) is d-good. (P5) X F := {ψ F Stab AG (s)(ψ) = Z G (u) F } =. Remark If G is adjoint then Properties (P2) and (P5) are trivially satisfied, so we will only need to concern ourselves with this when G has a disconnected centre. Furthermore, although strange in appearance Property (P5) is important for applications to generalised Gelfand Graev representations (as is explained in [Tay13]).

5 5 3. The Setup 3.1. We now introduce the appropriate notation and machinery that we will need to check Theorem 2.11 effectively. Firstly we fix an algebraic closure Q l of the field of l-adic numbers with l a prime distinct from p and we assume that any representation of a finite group is taken over Q l. Let us also note here that N = {0, 1, 2, 3,... } will denote the set of all natural numbers including 0. We assume fixed an isomorphism of groups ı : (Q/Z) p K and an injective homomorphism of groups j : Q/Z Q l. Note that (Q/Z) p is the subgroup of all elements whose order is coprime to p. The composition j ı 1 gives an injective homomorphism κ : K Q l. As G is simple we can and will express the Frobenius endomorphism F as a composition F r τ where τ is a graph automorphism of G and F r is a field automorphism of G for some r a power of p, (note that r = q if τ is trivial) Let us denote the root datum of G, (resp. G ), relative to T 0, (resp. T 0 ), by (X, Φ, q X, q Φ), (resp. (X, Φ, q X, q Φ )). Here X := X(T 0 ) = Hom(T 0, K ) and q X := q X(T 0 ) = Hom(K, T 0 ) are the sets of all algebraic group homomorphisms containing respectively the roots Φ and coroots q Φ of G, (similarly in the dual case). We assume Φ and q q Φ are the sets of simple roots and simple coroots determined by our choice of Borel subgroup B 0. For each α we denote by m α N the natural numbers such that α 0 = α m α α Φ is the unique lowest root of Φ, (this exists as G is simple). We will denote by = {α 0 } the set of extended simple roots and by S 0 = {s α α } the corresponding set of reflections. Assume is defined with respect to Φ in the same way as is defined with respect to. Then similarly we denote by T 0 = {t α α } the corresponding set of reflections in W. Denote by RX the R-vector space R Z X and by R q X the R-vector space R Z q X. These spaces have a canonical perfect pairing which we denote by, : RX R q X R. We denote by Ω = {ϖ α α } RX, (resp. q Ω = { qϖ α α } R q X), the basis dual to q, (resp. ) and define the fundamental group of the root system to be the quotient group Π = ZΩ/ZΦ, (similarly q Π = Z q Ω/Z q Φ is the dual fundamental group). We will assume that the extended set of roots is indexed as {α 0, α 1,..., α n }, where n is the semisimple rank of G, with the explicit labelling taken as in [Bou02, Plate I - IX]. For each 1 i n we denote respectively s αi, t αi ϖ αi, qϖ αi and m αi simply by s i, t i, ϖ i, qϖ i and m i. Following the conventions of Bonnafé we let qϖ α0 = 0 and m α0 = 1, (see [Bon05, 3.B]), we also denote s α0 S 0, (resp. t α0 T 0 ), simply by s 0, (resp. t 0 ) To G and G we fix simply connected covers δ sc : G sc G and δ sc : G ad G and adjoint quotients δ ad : G G ad and δ ad : G G sc, (note that G ad and G sc should be interpreted as (G ad ) and (G sc ) respectively). The kernels of these covers are simply the centres of the appropriate groups. If G is not of type B or C then we will have G ad and G sc, (resp. G sc and G ad ), are isomorphic as they are simply connected, (resp. adjoint), groups of the same type. Therefore we may and will take Gad = G sc and Gsc = G ad in this case. We will assume that the isogenies δ sc, δsc, δ ad and δad are chosen such that the compositions satisfy δ ad δ sc = δad δ sc. Note that such isogenies always exist as a consequence of the isogeny theorem for algebraic groups, (see for instance the remarks in [Tay12, 1.2]). By [MT11, Proposition 22.7] there exist Frobenius endomorphisms of G sc and G ad, which we again denote by F, such that δ sc and δ ad are defined over F q. Similarly we will denote by F a Frobenius endomorphism of both G sc and G ad such that δ sc and δ ad are also defined over F q Let us fix a maximal torus and Borel subgroup T sc B sc G sc then we may assume that T 0 B 0 are the images of T sc B sc under the isogeny δ sc. Furthermore we may define a maximal torus and

6 6 Borel subgroup T ad B ad G ad by letting these be the images of T sc B sc under the isogeny δ ad δ sc. Similarly we fix a maximal torus and Borel subgroup Tad B ad G ad and we assume T 0 B 0 are the images of Tad B ad under the isogeny δ sc. As before we define a maximal torus and Borel subgroup Tsc Bsc Gsc to be the images of Tad B ad under δ ad δ sc, (note once again that Tad and T sc should be interpreted as (T ad ) and (T sc ) respectively similarly for the Borel subgroups). If G is not of type B or C then we will assume T sc = Tad and B sc = Bad. We now assume that the duality isomorphisms ϕ ad, ϕ and ϕ sc are chosen such that the following diagram is commutative. X(T ad ) X(T 0 ) X(T sc ) ϕ ad ϕ ϕ sc qx(t ad ) q X(T 0 ) q X(T sc ) Here the horizontal maps are those induced by our chosen isogenies. The above maps give us bijections between the sets of roots and coroots so we will denote by Φ, q Φ the common set of roots and coroots of G ad, G and G sc ; also we write Φ, q Φ for the common set of roots and coroots of G sc, G and G ad. If G is not of type B or C then we will have X(T ad) = X(T sc) and qx(t ad ) = q X(T sc), which means we also have Φ = Φ and q Φ = q Φ. The Coxeter systems of G ad and G sc will be identified with (W, S) through the above isogenies. Similarly we will identify the Coxeter systems of G sc and G ad with (W, T) through the above isogenies. 4. Classes and Characters 4.1. We set out here the labelling conventions that will be maintained throughout. Assume H is a connected reductive algebraic group whose derived subgroup H is simple. Assume that H is of classical type then the elements of Cl U (H) will be described in terms of partitions as in [Car93, 13.1]. Specifically the partition of O Cl U (H) is given by the elementary divisors of u O in a natural matrix representation of H. If H is of exceptional type then the elements of Cl U (H) are described using the Bala Carter labelling, which is also described in [Car93, 13.1]. For the parameterisation of the characters of Weyl groups, (except for the case of G 2 ), and unipotent characters we will follow the parameterisation defined in [Lus84a, Chapter 4]. In particular if W is a Weyl group of type B n, (resp. D n ), then Irr(W) will be parameterised in terms of symbols of rank n and defect 1, (resp. defect 0). However, for notational convenience, we will denote the two-row symbol [ ] A B by [A; B]. In the case of Weyl groups of type G2 we will follow the labelling given in [Car93, 13.2]. It is clear from Theorem 2.11 that we will also need to know part of the Springer correspondence. The image of the springer correspondence contains the subset {(O, Q l ) O Cl U (H)} of N H and it is this part of the map that will interest us. If H is of classical type then this part of the Springer correspondence is described combinatorially in [GM00, 2]. If H is of exceptional type then the Springer correspondence is given by the tables in [Car93, 13.3]. We will denote by ρ(o) Irr(W) the character corresponding to the pair (O, Q l ) under the Springer correspondence. In type D n all of the labelling sets considered above have some ambiguity with regard to degenerate elements. Here we say a unipotent class O Cl U (G) is degenerate if it is parameterised by a partition λ 2n all of whose entries are even. We say a character of W is degenerate if its corresponding symbol is degenerate, (in the sense of [Lus84a, 4.6]). We will use a ± notation to distinguish between all degenerate

7 7 elements, note that in [Lus84a, Chapter 4] Lusztig uses the notation s /s. We will make this concrete below by describing explicitly the Springer correspondence in this case To prove Theorem 2.11 we will need to be able to discern the action of automorphisms on unipotent characters. In this direction we have the following result which was already known to Lusztig in [Lus88] but was formalised by Digne Michel in [DM90, Proposition 6.6] and Malle in [Mal91, 1], (see also [Mal07, Proposition 3.7]). Lemma 4.3. Assume H is simple, F : H H is a Frobenius endomorphism and γ is an automorphism of H commuting with F. If γ induces the identity on W then every unipotent character of H is fixed under composition with γ. Assume H is of type A n, D n or E 6 and γ acts on the corresponding Coxeter system (W, S ) as a non-trivial graph automorphism. Then every unipotent character is fixed under composition with γ except in the following cases: H is of type D 2n, γ has order 2 and the character is parameterised by a degenerate symbol. H is of type D 4, γ has order 3 and the character is parameterised by one of the symbols [2; 2] ± [12; 12] ± [01; 14] [012; 124]. (4.4) Remark 4.5. Recall that we will be interested in the action of A G (s) by conjugation on the unipotent characters of C G (s) for some semisimple element s T0. Note that each such automorphism is of the form stated in Lemma Explicit Descriptions for Half-Spin Groups When dealing with the simple algebraic groups of type D n, with n 4, we must be quite careful. In this section we assume G is such a group. Note that the notational conventions we develop in this section for such groups shall be maintained throughout. Describing Half-Spin Groups 5.1. We start by considering precisely the structure of the fundamental group Π = ZΩ/ZΦ. From [Bou02, Plate IV(VIII)] we have the fundamental group is given by Π = {ZΦ, ϖ 1 + ZΦ, ϖ n 1 + ZΦ, ϖ n + ZΦ}, which is isomorphic to C 2 C 2 if n 0 (mod 2) and C 4 if n 1 (mod 2), (here C m is a cyclic group of order m). Recall that as G is semisimple we have the isomorphism type of G is determined by the image of X in the fundamental group. If the image of X is the subgroup generated by ϖ 1 + ZΦ then G is a special orthogonal group SO 2n (K). If n 0 (mod 2) then there are two other non-trivial cases, namely if the image of X is the subgroup generated by ϖ n 1 + ZΦ or ϖ n + ZΦ then G is a half-spin group HSpin 2n (K), (see for instance [Car81, 7]). The problem arises here in the choice over the root datum of a half-spin group. Note these groups are isomorphic because there exists an isomorphism of their root data which exchanges the weights ϖ n 1 and ϖ n. We will now fix a choice of half-spin group but it will be clear, because of this isomorphism, that the results we prove do not depend upon this choice. If G is a half-spin group then we assume the image of X in the fundamental group Π is ϖ n + ZΦ.

8 Let us assume now that G is a half-spin group and fix a basis {χ 1,..., χ n } of RX such that χ 1 = ϖ n and χ i = α i for 2 i n. We write A for the change of basis matrix of RX sending the simple roots to {χ 1,..., χ n }. This matrix has the form a 1 a 2 a n 0 A =. I n 1, 0 where (a 1,..., a n ) = ( 1 2, 1, 3 (n 2) 2, 2,..., 2, (n 2) 4, n 4 ) and I n 1 is the (n 1) (n 1) identity matrix. To our chosen basis {χ 1,..., χ n } of RX we have a dual basis {γ 1,..., γ n } of RX. q Let B be the change of basis matrix of R q X sending the simple coroots q to {γ 1,..., γ n }. The matrices A and B satisfy the condition ACB T = I n where C = ( α i, qα j ) 1 i,j n is the Cartan matrix and I n is the n n identity matrix. Let us now consider the root datum of the associated dual group G. If X has image ϖ n + ZΦ in Π then for X to be isomorphic to q X, (and for such an isomorphism to preserve the pairing, ), we must have q X has image qϖ n + Z q Φ in q Π. From this we can easily calculate the image of q X in q Π as follows. Recall that the matrix B expresses the decomposition of the basis of R q X in terms of the simple coroots. The Cartan matrix expresses the decomposition of the simple coroots in terms of the fundamental dominant coweights hence BC = A T gives the decomposition of {γ 1,..., γ n } in terms of { qϖ 1,..., qϖ n }. We easily determine that this matrix has the form a A T a = 2. I n 1, a n where (a 1,..., a n) = (2, 2, 3,..., (n 2), (n 2) 2, n 2 ). From this we see that the image of q X in q Π is determined by the image of γ n in q Π, i.e. the element n 2 q ϖ 1 + qϖ n + Z q Φ. Therefore we have the image of q X in q Π is qϖ n + Z q Φ if n 0 (mod 4) and qϖ n 1 + Z q Φ if n 2 (mod 4). In particular, using the dual isomorphism q X X, we must have the image of X in Π is ϖ n + ZΦ if n 0 (mod 4) and ϖ n 1 + ZΦ if n 2 (mod 4). As a consequence we see that the dual group of a half-spin group is again isomorphic to a half-spin group but its root datum depends upon n. It will be useful for us to also describe Ker(δ sc) but to do this we need the following lemma. Lemma 5.3 (see [Bon06, Proposition 4.1]). Let G be a connected semisimple algebraic group. There exists a canonical surjective homomorphism Q Z q X(T0 ) T 0 which induces an isomorphism (Z q Ω/ q X) p Z(G). Note that the above isomorphism depends upon the choice of ı. Taking the above lemma in the case where G is simply connected of type D n this says we have a natural isomorphism Π q Z(G), (recall we assume p = 2). We now make the following convention regardless of the congruence of n (mod 2). Assume G is a simply connected group of type D n then we denote the centre of G by Z(G) = {1, ẑ 1, ẑ n 1, ẑ n }. We fix the notation such that qϖ n 1 + ZΦ q ẑ n 1 and qϖ n + ZΦ q ẑ n under the isomorphism specified by Lemma 5.3.

9 9 Under this convention whenever G is a half-spin group we will have Ker(δsc) = ẑ n, however Ker(δ sc ) will be ẑ n if n 0 (mod 4) and ẑ n 1 if n 2 (mod 4). Furthermore whenever G is a special orthogonal group we will have Ker(δsc) = Ker(δ sc ) = ẑ 1. The Springer Correspondence 5.4. In this section we wish to remove the ambiguity over the labelling of elements in Cl U (G) and Irr(W) and in particular express concretely the Springer correspondence for degenerate elements. Recall that for this situation to occur we must necessarily have n 0 (mod 2). To clarify the Springer correspondence we will use the argument given in [Car93, 13.3]. Let λ 2n be a degenerate partition then we can express λ as (2η 1, 2η 1,..., 2η s, 2η s ), where s, η i N. We denote by η the sequence (η 1,..., η s ) then η is a partition of n/2. If ξ = (ξ 1,..., ξ r ) n/2 is a partition of n/2 then there are two G-conjugacy classes of Levi subgroups with semisimple type A 2ξ1 1 A 2ξr 1. Two class representatives can be given by standard Levi subgroups and the two possibilities depend on whether the root α n or α n 1 is contained in the root system of the standard Levi. We will denote by L + ξ the Levi subgroup whose root system contains α n 1 and L ξ the Levi subgroup whose root system contains α n. By the Bala Carter theorem, (see [Car93, Theorem 5.9.5]), if O Cl U (G) is a unipotent class parameterised by the degenerate partition λ then either O L + η contains the regular unipotent class of L + η or O L η contains the regular unipotent class of L η. We now turn to the irreducible characters of W. Assume [Λ] ± Irr(W) are the two irreducible characters parameterised by the degenerate symbol [Λ]. Adopting the notation above we denote the Weyl groups of the standard Levi subgroups L ± ξ by W(A± ξ ). By [GP00, Theorem 5.4.5] and [GP00, Proposition 5.6.3] there is a unique partition ξ n/2 such that {[Λ] +, [Λ] } = {j W W(A + ξ ) (sgn), jw (sgn)} W(A where ξ ) sgn Irr(W(A ± ξ )) denotes the sign character and ξ denotes the dual partition. With this we now distinguish the degenerate objects in the following way. We assume the ± convention to be chosen such that [Λ] ± = j W (sgn) W(A ± for some ξ ) (unique) partition ξ n/2. Furthermore O ± λ Cl U(G) is the (unique) unipotent class such that O ± λ L± η contains the regular unipotent class of L ± η, with η as above We now come to an interesting dichotomy, (which is the duality discussed by Spaltenstein in [Spa82, Chapitre III]). The way we have identified the two degenerate unipotent classes will allow us to compute the order of the component groups of their centraliser, however it will not allow us to compute the Springer correspondence. For this we must identify these classes as Richardson classes associated to their canonical parabolic subgroup. Let us denote by η the partition of n/2 dual to η then we have the following result. Proposition 5.6 (see [Spa82, Proposition II.7.6]). The unipotent classes O ± λ are Richardson classes for parabolic subgroups with Levi complement L ± η if n 0 (mod 4) and L η if n 2 (mod 4). Furthermore, let us denote by (n ± α n 1, n ± α n ) the weights of the weighted Dynkin diagram of O ± λ associated to the nodes α n 1 and α n. Then we have (n + α n 1, n + α n ) = (a, b) and (n α n 1, n α n ) = (b, a) where b = 2 a and 0 if n 0 (mod 4), a = 2 if n 2 (mod 4).

10 10 Remark 5.7. The statement concerning the weighted Dynkin diagram follows from [Spa82, Proposition II.7.6] because the classes are even so they are Richardson classes for their canonical parabolic subgroups, (see [Hum95, Proposition]) Assume L is a Levi subgroup of G contained in a parabolic subgroup P with unipotent radical U P. In [LS79] Lusztig and Spaltenstein have defined an induction map Ind G L taking a unipotent conjugacy class of L to a unipotent conjugacy class of G, which is defined in the following way. If O is a unipotent class of L then Ind G L (O) is the unique unipotent conjugacy class of G such that Ind G L (O) OU P is dense in OU P. They show that this does not depend on the choice of P and depends only on the pair (L, O) up to G conjugacy. Hence we may assume that L is a standard Levi subgroup of G. Note that the statements in [LS79] have some restrictions but these were removed in [Lus84b]. Proposition 5.9 (Lusztig and Spaltenstein, [LS79, Theorem 3.5]). Assume O is a unipotent conjugacy class of a Levi subgroup L of G containing T 0. Write ρ(o) Irr(W(L)) for the character corresponding to (O, Q l ) under the Springer correspondence of L. Let Õ = Ind G L (O) be the induced class and write ρ(õ) Irr(W) for the character corresponding to (Õ, Q l ) under the Springer correspondence of G then ρ(õ) = j W W(L) (ρ(o)) Using Proposition 5.6 and [LS79, Proposition 1.9(b)] we have O ± λ is IndG L ± η (O 0) if n 0 (mod 4) and Ind G L η (O 0) if n 2 (mod 4), where O 0 denotes the trivial unipotent class. Note that there is a restriction on [LS79, Proposition 19.(b)] that p is sufficiently large but this is only to ensure that unipotent classes are parameterised by their weighted Dynkin diagrams, which is known to hold in good characteristic. The Springer character of the trivial class is always the sign character, therefore we have j W (sgn) ρ(o ± λ ) = W(A ± if n 0 (mod 4), η ) j W W(A η ) (sgn) if n (5.11) 2 (mod 4). This now concretely specifies the Springer correspondence in the degenerate case. Component Groups in Half-Spin Groups We now come to the determination of A G (u ± ) where u ± O ± λ are class representatives for degenerate unipotent classes. From the description of the component groups given in [Lus84b, 14.3], [Lus84b, 10.6] and [Car93, 13.1] we have A G (u + ) = A G (u ) except when G is a half-spin group, in which case we always have A G (u + ) = A G (u ). Assume G is a half-spin group then we claim that A G (u + 2 if n 0 (mod 4), ) = 1 if n 2 (mod 4), A G (u 1 if n 0 (mod 4), ) = 2 if n 2 (mod 4), Let us now verify this claim. Firstly let u ± ad = δ ad(u ± ) be corresponding elements in the adjoint group then we have A Gad (u ± ad ) = 1, (see [Car93, 13.1]). In particular we must have A G(u ± ) = Z G (u ± ) hence A G (u ± ) {1, 2} depending upon Z G (u ± ). The intersection O ± λ L± η contains the regular unipotent class of L ± η therefore we may take u ± O ± λ to be such that it is a regular unipotent element of L ± η. We have a natural embedding C L ± η (u ± ) C G (u ± ), which induces an embedding A L ± η (u ± ) A G (u ± ). As u ± is a regular unipotent element we

11 11 have A L ± η (u ± ) = Z(L ± η ) where Z(L ± η ) denotes the component group Z(L ± η )/Z(L ± η ), (see for example the proof of [DM91, Proposition 14.24]). Hence to determine whether A G (u ± ) = 2 or 1 it is enough to determine when Z(L ± η ) = 2 or 1. To do this calculation we will use a result of Digne Lehrer Michel. Recall that we have a natural embedding Z(G) Z(L ± η ), which induces a surjective map Z(G) Z(L ± η ) by [Bon06, Proposition 4.2]. The kernel of this surjective map is given to us by the following result. Proposition 5.13 (Digne Lehrer Michel, [Bon06, Proposition 4.5]). Let I be a set of simple roots and L I the standard Levi subgroup corresponding to I. The kernel of the map Z(G) Z(L I ) is the image of qϖ α + q X α \ I under the isomorphism (Z q Ω/ q X) p = Z(G) of Lemma Recall from Section 5 that the image of q X in q Π depends upon the congruence of n (mod 4). We treat the two cases separately. n 0 (mod 4) then q X = qϖ n + Z q Φ. By Proposition 5.13 we have the kernel of the map Z(G) Z(L ± η ) is non-trivial whenever α n 1 is not in the root system of the Levi. If the kernel is non-trivial then the order of Z(L ± η ) is 1. Hence we have Z(L + η ) = 2 and Z(L η ) = 1. n 2 (mod 4) then q X = qϖ n 1 + Z q Φ. By Proposition 5.13 we have the kernel of the map Z(G) Z(L ± η ) is non-trivial whenever α n is not in the root system of the Levi. If the kernel is non-trivial then the order of Z(L ± η ) is 1. Hence we have Z(L + η ) = 1 and Z(L η ) = 2. This now verifies the statements regarding the component group orders. 6. Quasi-Isolated Semisimple Elements The results we prove in this section will be stated in terms of G, for notational convenience, but they will be applied to the dual group. The exception to this will be in 6.7 to 6.13 and 6.19 where we prove results concerning the relationship between G and G. Bonnafé s Classification 6.1. We start by describing Bonnafé s classification of quasi-isolated semisimple elements, (see [Bon05]). Assume T G is a maximal torus and recall that we have an isomorphism K Z X(T) q T given by k γ γ(k). Using the isomorphism ı : (Q/Z) p K we obtain an isomorphism ı T : (Q/Z) p Z qx(t) T given by ı T (r γ) = γ(ı(r)). We have an action of F on (Q/Z) p Z X(T) q given by F(r γ) = r F(γ) which is compatible with the action of F on T. As G ad is adjoint the cocharacter group q X(T ad ) can be identified with the coweight lattice, which means we can naturally consider all fundamental dominant coweights qϖ α q Ω to be elements of q X(T ad ). Let A := Aut W ( ) = {x W x( ) = } W be the automorphism group of the extended Dynkin diagram in W and let Q(G ad ) denote the set of subsets Σ such that the stabiliser of Σ in A acts transitively on Σ. We then have the following theorem of Bonnafé, (recall that we assume here that p is a good prime for G). Theorem 6.2 (Bonnafé, [Bon05, Theorem 5.1]). Let Σ Q(G ad ) and define an element t Σ T ad by setting ( ) 1 t Σ = ı Tad α Σ m α Σ qϖ α,

12 12 G ad Σ m α Σ C Gad (t Σ ) A Gad (t Σ ) Isolated? A n {α j(n+1)/d 0 j d 1} d n+1 and p d d (A (n+1 d)/d ) d d d = 1 B n {α 0, α 1 } 2 B n 1 2 no {α 0 } 1 B n 1 yes {α d }, d [2, n] 2 D d B n d 2 yes C n D n {α 0 } 1 C n 1 yes {α d }, d [1, n 1] \ {n/2} 2 C d C n d 1 yes {α n/2 }, (only if 2 n) 2 C n/2 C n/2 2 yes {α 0, α n } 2 A n 1 2 no {α d, α n d }, 1 d < n/2 4 C d A n 2d 1 C d 2 no {α 0 } 1 D n 1 yes {α d }, d [2, n 2] \ {n/2} 2 D d D n d 2 yes {α n/2 } (only if 2 n) 2 D n/2 D n/2 4 yes {α d, α n d }, d [2, n 2] \ {n/2} 4 D d A n 2d 1 D d 4 no {α 0, α 1, α n 1, α n } 4 A n 3 4 no {α 0, α 1 } 2 D n 1 2 no {α 0, α n 1 }, (only if 2 n) 2 A n 1 2 no {α 0, α n }, (only if 2 n) 2 A n 1 2 no Table 6.1: Classes of Quasi-Isolated Semisimple Elements in Classical Groups where qϖ α Ω, q (and m α is as in 3.2). The following then hold: the map Σ t Σ induces a bijection between the set of orbits of A acting on Q(G ad ) and the set of conjugacy classes of quasi-isolated semisimple elements in G ad. for any Σ Q(G ad ) we have: W(t Σ ) = s α S 0 α Σ ; A Gad (t Σ ) = {xw(t Σ ) x A and x(σ) = Σ}. Remark 6.3. In the statement of the above theorem we have identified A G (s) and W(s)/W(s) under the usual natural isomorphism between these two groups, (see for instance [Bon05, Proposition 1.3(d)]). We will maintain this identification throughout An important aspect of Bonnafé s theorem is that he determines the structure of A G (t Σ ), which is important to us in verifying the validity of Property (P2). In Tables 6.1 and 6.2 we reproduce Bonnafé s classification of quasi-isolated semisimple elements in classical and exceptional adjoint algebraic groups, as found in [Bon05, Tables 2 and 3]. In the case of G 2, F 4 and E 8 the notion of isolated and quasi-isolated semisimple elements coincide as the adjoint and simply connected groups coincide. Note that in the original table of Bonnafé the class representative for the class corresponding to {α n/2 } in D n is denoted as having m α Σ = 4. However it is clear that this element has m α Σ = 2 as it is isolated, (see [Bon05, Proposition 5.5]).

13 13 G ad Σ C Gad (t Σ ) A G (t Σ ) Isolated? G 2 {α 1 } A 1 A 1 1 yes {α 0 } G 2 1 yes {α 2 } A 2 1 yes F 4 E 6 E 7 E 8 {α 0 } F 4 1 yes {α 1 } A 1 C 3 1 yes {α 2 } A 2 A 2 1 yes {α 3 } A 3 A 1 1 yes {α 4 } B 4 1 yes {α 0 } E 6 1 yes {α 2 } A 5 A 1 1 yes {α 4 } A 2 A 2 A 2 3 yes {α 0, α 1, α 6 } D 4 3 no {α 2, α 3, α 5 } A 1 A 1 A 1 A 1 3 no {α 0 } E 7 1 yes {α 1 } A 1 D 6 1 yes {α 2 } A 7 2 yes {α 3 } A 2 A 5 1 yes {α 4 } A 3 A 3 A 1 2 yes {α 0, α 7 } E 6 2 no {α 1, α 6 } D 4 A 1 A 1 2 no {α 3, α 5 } A 2 A 2 A 2 2 no {α 0 } E 8 1 yes {α 1 } D 8 1 yes {α 2 } A 8 1 yes {α 3 } A 1 A 7 1 yes {α 4 } A 2 A 1 A 5 1 yes {α 5 } A 4 A 4 1 yes {α 6 } D 5 A 3 1 yes {α 7 } E 6 A 2 1 yes {α 8 } E 7 A 1 1 yes Table 6.2: Classes of Quasi-Isolated Semisimple Elements in Exceptional Groups The Group A 6.5. We will need to know explicitly the exact structure and actions of the group A G (t Σ ). To do this we will need to describe explicitly the group A. If G is not of type D n then A is cyclic and is described in [Bou02, Plates I-IX(XII)]. To fix the notation in the case of type D n we recall the description of A from [Bou02, Plate IV(XII)]. We denote the elements of A by the set {1, σ 1, σ n 1, σ n }. If n 0 (mod 2) then the element σ n 1 acts by exchanging the elements in the sets {α 0, α n 1 }, {α 1, α n }, {α j, α n j } for each 2 j n 2. The element σ n acts by exchanging α j with α n j for all 0 j n. Furthermore A is generated by σ n 1 and σ n. If n 1 (mod 2) then the element σ n acts by mapping α 0 α n α 1 α n 1 α 0 and exchanges α j with α n j for 2 j n 2. Furthermore A is generated by σ n. The element σ 1 always acts by exchanging the elements in the sets {α 0, α 1 }, {α n 1, α n } and fixes α j for all 2 j n Recall from [Bon05, 3.7] that we have an isomorphism A q Π. If G is simply connected and

14 14 qπ p = q Π, (i.e. p is a very good prime for G), then by Lemma 5.3 we also have an isomorphism q Π Z(G). By composing these isomorphisms we have an isomorphism A Z(G). We wish to describe this isomorphism in the case where G is of type D. We describe the isomorphism A q Π following [Bon05, 3.B]. Let α j be a root for some j then we denote by j the set \ {α j }. We write Φ j Φ for the parabolic subsystem generated by the set j and W j = s α α j the corresponding parabolic subgroup of W. Let Φ + j = Φ j Φ + be a system of positive roots for Φ j then we denote by w j W j the unique element such that w j (Φ + j ) = Φ + j, (i.e. the longest word in W j ). Define x j = w j w 0 W then A = {x j m αj = 1} by [Bon05, 3.5]. The isomorphism A q Π is then given by x j qϖ j + Z q Φ. We consider what this means for a simply connected group of type D n. In this case we have A = {x 0, x 1, x n 1, x n }, (see [Bon05, Table 1]). It is clear from the description that x 0 is the identity. If j is n 1 or n then it is easy to determine the action of x j because the longest word in W j will induce the unique non-trivial graph automorphism on the root system j of type A n 1. Furthermore the longest element w 0 W will induce no graph automorphism if n is even and the unique graph automorphism of order 2 if n is odd. Comparing with Section 5 we see that we have chosen the labelling such that σ j ẑ j, (for j {1, n 1, n}), under the isomorphism A Z(G). Component Groups of Semisimple Elements 6.7. In this section we will be interested in A G (s) where s T0 is a semisimple element such that s = δad (s) T sc is quasi-isolated. An expression for this is already obtained by Bonnafé in [Bon05, Proposition 3.14(b)], however we wish to determine a slightly different description which will make numerical comparisons between component groups of unipotent elements in G simpler. If G is simply connected then we know that the centraliser of every semisimple element is connected, (by the classical work of Steinberg), so this value will always be 1. It suffices therefore to only consider simple groups which are neither simply connected nor adjoint, (as the adjoint case is dealt with in Tables 6.1 and 6.2). These groups can only occur in types A n and D n so we may assume that G is such a simple group. Let ŝ Tad be such that δ sc(ŝ) = s T0. Following Bonnafé [Bon05, 2.B] we define two homomorphisms ω s : C G sc ( s) Z(Gad ) and ω s : C G (s) Z(Gad ) by setting ω s( x) = [ŝ, ˆx] and ω s (y) = [ŝ, ŷ], where ˆx, ŷ are such that (δ ad δ sc)( ˆx) = x and δ sc(ŷ) = y. We recall the following result of Bonnafé. Lemma 6.8 (Bonnafé, [Bon05, Corollary 2.8]). The homomorphisms ω s, ω s induce embeddings ω s : A G sc ( s) Z(Gad ) and ω s : A G (s) Z(Gad ). Their respective images are given by Im( ω s ) = {ẑ Z(G ad ) ŝ and ŝẑ are conjugate in G ad }, Im( ω s ) = {ẑ Ker(δ sc) ŝ and ŝẑ are conjugate in G ad }. It is easily checked that we have ω s F s = F ω s and ω s F s = F ω s. From this lemma we see that A G sc ( s) = Im( ω s ) and A G (s) = Im( ω s ) so to determine A G (s) we need only determine Im( ω s ) Ker(δ sc). In later sections we will want to compare A G (s) with A G (u) for some unipotent element u G. We now take the time to prove some small results which will facilitate this. Lemma 6.9. The groups Ker(δ sc ) and Irr(Z(G )) are isomorphic and this isomorphism is defined over F q.

15 15 Proof. The restriction of the isogeny δ sc to the maximal tori T sc T 0 gives rise to an injective homomorphism q X(T sc ) q X(T 0 ). By [Bon06, Proposition 1.11] this induces an isomorphism ( q X(T 0 )/ q X(T sc )) p = Ker(δ sc ), where we identify q X(T sc ) with its image in q X(T 0 ). Using duality this gives rise to an isomorphism (X(T 0)/X(T sc)) p = ( q X(T0 )/ q X(T sc )) p = Ker(δsc ). Recall that X(T sc) can be identified with ZΦ so by [Bon06, Proposition 4.1] we have a natural isomorphism X(Z(G)) = (X(T 0 )/X(T sc)) p. The morphism X(Z(G )) Irr(Z(G )), given by χ κ χ is an isomorphism of finite abelian groups. Finally, checking the statements in [Bon06], we can see that all morphisms are defined over F q. Corollary We have Ker(δ sc ) F = Z(G ) F. Proof. From Lemma 6.9 we see that Ker(δ sc ) F = Irr(Z(G )) F because the isomorphism is defined over F q. The group Irr(Z(G )) F is canonically isomorphic to the group Irr(H 1 (F, Z(G ))), (here H 1 (F, Z(G )) is defined as in [Bon06, 1.B]), because χ is an element of Irr(Z(G )) F if and only if χ(f (z)) = χ(z) for all z Z(G ). On the other hand this is true if and only if χ(z 1 F (z)) = χ(1) for all z Z(G ), which means Ker(χ) = (F 1)Z(G ). Using, for instance [Bon05, Exemple 1.1], we see that Ker(δ sc ) F = Irr(Z(G )) F = H 1 (F, Z(G )) = Z(G ) F. To obtain the second equality we have used the fact that H 1 (F, Z(G )) has the same order as its character group because it is a finite abelian group. We finally end this discussion on component groups with a particularly useful observation relating fixed point groups. Lemma Assume G has a cyclic centre and A Ker(δ sc) and Z Z(G) are subgroups of common order d = A = Z then A F = Z F. Proof. By Lemma 6.9 we have Ker(δ sc) = Z(G) = N, hence these groups are isomorphic to a cyclic group of order N. We may assume that 1 n, m N are such that F (x) = x n and F(y) = y m, where Ker(δ sc) = x and Z(G) = y. Taking λ = N/d it is easily seen that A F = λ gcd(n 1, d) and Z F = λ gcd(m 1, d). To show that these groups have the same order it is enough to show that gcd(n 1, d) = gcd(m 1, d). However, because gcd(n 1, d) = gcd(n 1, d, N) = gcd(n 1, N), (similarly for gcd(m 1, d)), it is sufficient to show that gcd(n 1, N) = gcd(m 1, N) but this is just a restatement of Corollary Groups of Type A n If G is a group of type A n then Z(Gad ) is a cyclic group so this simplifies trying to understand A G (s). We know A G sc ( s) = Im( ω s ) in particular, as Ker(δsc) is cyclic, we know z Im( ω s ) Ker(δsc) if and only if the order of z divides Im( ω s ) and Ker(δ sc). Hence it is easy to see that we have A G (s) = gcd( A G sc ( s), Ker(δ sc) ).

16 16 Assume u G is a unipotent element and u sc is the unique unipotent element in δ 1 sc (u). Using the natural exact sequence Ker(δ sc ) A Gsc (u sc ) A G (u) 1 we have A G (u) = A Gsc (u sc ) / gcd( Ker(δ sc ), A Gsc (u sc ) ). Let d := A G (u), which is a divisor of A Gsc (u sc ). By the description of A Gsc (u sc ) given in [Lus84b, 10.3] we have d is a divisor of n + 1 and p d. Therefore there exists a semisimple element s T0 such that s = δad (s) is quasi-isolated and A Gsc ( s) = d. As A G(u) = Z G (u) we have d divides Z(G) = Ker(δsc) so A G (s) = d. What we have shown here is that for each unipotent conjugacy class O of G there exists a semisimple element s T 0 such that A G(u) = A G (s). Groups of Type D n If G is a simple group of type D n, which is neither simply connected nor adjoint, such that n 1 (mod 2) then G must be a special orthogonal group. The kernel Ker(δ sc) will be the unique subgroup of order 2 in Z(G sc ) so 1 if A Gad ( s) = 1, A G (s) = 2 if A Gad ( s) 2. If n 0 (mod 2) then G is either isomorphic to a special orthogonal group or a half-spin group. It is clear that if A Gad ( s) = 1 or 4 then we will respectively have A G (s) = 1 or 2. The problem now arises when A Gad ( s) = 2. Assume s is a quasi-isolated semisimple element with this property then in Table 6.3 we describe the orders of A G (s) depending upon whether G is a special orthogonal group or a half-spin group. To determine the information in Table 6.3 one has to only check which element of A stabilises Σ then see if the corresponding element of Z(G sc ) lies in Ker(δ sc), (using the description in 6.6). Please Note: from the discussion in Section 5 we always have Ker(δ sc) = ẑ n when G is a half-spin group, chosen as in Section 5. Hence the information in Table 6.3 holds regardless of the congruence of n (mod 4). Σ C G ad ( s) SO 2n (K) HSpin 2n (K) {α d }, 2 d < n/2 D d D n d 2 1 {α n/2 }, (only if 2 n) D n/2 D n/2 2 2 {α 0, α 1 } D n {α 0, α n 1 } A n {α 0, α n } A n Table 6.3: Component Group Orders in Groups of Type D n F-stability of Classes in Adjoint Groups In the remainder of this section we wish to address two issues. Firstly we wish to show that, modulo some exceptions, every class of quasi-isolated semisimple elements in a simple adjoint algebraic group is F-stable. Secondly we wish to show that, if C is an F-stable class of quasi-isolated semisimple elements of G ad then there exists an F-stable class of G sc whose image under δ ad δ sc is C.

17 17 Proposition Let G ad be a simple adjoint algebraic group of classical type and F a Frobenius endomorphism written as F r τ where τ is a graph automorphism of G ad. Given any set Σ Q(G ad ) we have t Σ is conjugate to F(t Σ ) unless: G ad is of type D n, the graph automorphism τ is of order 2 and Σ is {α 0, α n 1 } or {α 0, α n }. G ad is of type D 4, the graph automorphism τ is of order 3 and Σ = {α 0, α 1 }, {α 0, α 3 } or {α 0, α 4 }. In particular, except for those mentioned above, the conjugacy class containing t Σ is F-stable. Proof. Let C Σ be the class of quasi-isolated semisimple elements of G ad such that t Σ C Σ. Furthermore let y Σ = ı 1 T ad (t Σ ) = α Σ 1/m α Σ qϖ α (Q/Z) p q X(T ad ) and recall that m α is constant on Σ. Using the fact that the tensor product is taken over Z we have the action of F on y Σ is given by F(y Σ ) = α Σ r/m α Σ qϖ τ(α). W-orbit. To show C Σ is F-stable we need only show that y Σ and F(y Σ ) lie in the same If Σ = {α 0 } then y Σ = 0, which is always F-stable. Let G ad be of type A n and assume τ is trivial. In this instance it will be much more transparent to work with a concrete realisation of G ad, namely PGL n+1 (K). Let d be a divisor of n + 1 and Σ the corresponding subset of the roots. Following Bonnafé we define a matrix J d = diag(1, η d, ηd 2,..., ηd 1 d ) GL d (K), where η d is a primitive dth root of unity in K. Let s Σ = I n+1/d J d GL n+1 (K) be the Kronecker product of the matrices, where I n+1/d GL n+1/d (K) is the identity matrix. Considering the standard quotient map π : GL n+1 (K) PGL n+1 (K) we have s Σ = π( s Σ ) is a representative in PGL n+1 (K) of the class parameterised by Σ. The action of the Frobenius is given by F( s Σ ) = I n+1/d diag(1, η q d, η2q d,..., ηq(d 1) d ). As q and d are coprime we have η q d = ηi d for some 1 i d 1 so the entries η q d,..., ηq(d 1) d are just a permutation of η d,..., η d 1 d. There is clearly an element of the Weyl group w d W such that F( s Σ ) = s w d If C Σ is the conjugacy class of GL n+1 (K) containing s Σ then π( C Σ ) = C Σ and F( C Σ ) = C Σ. π(f( C Σ )) = π( C Σ ) F(π( C Σ )) = F(C Σ ) = C Σ so C Σ is an F-stable class. Σ. As π is defined over F q we have Let G ad be of type A n and assume τ is of order 2. The map τ acts on the simple roots by sending α k α n+1 k for all 1 k n. Furthermore it is such that τ(α 0 ) = α 0. The roots in Σ are all of the form α j(n+1)/d for some 0 j d 1, where d is a divisor of n + 1 as in the previous case. If j = 0 then we have τ(α j(n+1)/d ) = α (d j)(n+1)/d so it is clear that τ preserves the set Σ. Hence y Σ and F(y Σ ) are in the same W-orbit. Let G ad be of type B n, C n or D n and assume τ is trivial. If m α Σ = 2 then F(y Σ ) = y Σ because q is odd hence q/2 = 1/2 Q/Z. Assume m α Σ = 4 then G ad must be of type C n or D n. If q 1 (mod 4) then q/4 = 1/4 Q/Z and F(y Σ ) = y Σ. If q 3 (mod 4) then q/4 = 3/4 = 1/4 Q/Z so F(y Σ ) = y Σ. If G ad is of type C n or it is of type D n and n 0 (mod 2) then the longest element w 0 W acts on the coweights by 1, (see [Bou02, Plates II - IV (XI)]), therefore F(y Σ ) and y Σ are conjugate by w 0. If G ad is of type D n and n 1 (mod 2) then the longest element w 0 W acts on the coweights as ε where ε is such that ε( qϖ i ) = qϖ i for all 1 i n 2 and ε exchanges qϖ n 1 and qϖ n, (see [Bou02, Plate IV (XI)]). All subsets Σ considered here are stable under ε so F(y Σ ) and y Σ are conjugate by w 0.